The slope is m in the equation y = mx + b
Remain constant if the x value is constant too
Parallel lines will have the same slope even is the y-intercept is different
The y-intercept is the b in the equation of the line
We can convert one into the other
That is why the the line in that graph is negative
Formula of the Slope
After finding the point using slope, we can create an equation
Partial Variation has direct variation in it, but its not the other way round
y-intercept form and Standard form are 2 ways to express an equation of a line
This is the same mathematical relation in word form
Slope can be found using the intercepts

Modelling with Graphs and Analyzing Linear Relations

The Equation of a Line in Standard form: Ax+ By+ C = 0

The equation of a line can be expressed in different ways. The second way that we can express the equation of a line is standard form.
It is expressed as Ax+ By+ C = 0 where A, B, and C are not fractions and A is positive. The equations 3x + 2 -5=0 and 2x -5y +7=0 are two examples of linear relations which are in standard form.

You can convert an equation from standard form to slope y-intercept form and vice versa.

x + 2y - 4 = 0 2y = -x + 4 2y/2 = -x +4/2 y = -1x/2 + 4/2 y = - ½x + 2

x + 2y - 4 = 0
2y = -x + 4
2y/2 = -x +4/2
y = -1x/2 + 4/2
y = - ½x + 2

1. Subtract x from both sides and add 4 to both sides.
2. Divide both sides by 2
3. Divide each term on the right side by 2

y = y = -3x + 5 3x + y - 5 = 0 Get all the values on the left side by doing opposite operation.

y = y = -3x + 5
3x + y - 5 = 0
Get all the values on the left side by doing opposite operation.

Identify fixed and variable costs in a partial variation

25n - C + 1250 = 0 -C = 25n - 1250 C/-1 = -25n/-1 = C C = 25n +

25n - C + 1250 = 0
-C = 25n - 1250
C/-1 = -25n/-1 = C
C = 25n + 1250

The Equation of a Line in a Slope y-intercept Form: y = mx+ b

The equation of a line can be written in slope y-intercept form: y = mx + b , where m is the slope of the line and b is the y

The equation of a line can be written in slope y-intercept form: y = mx + b , where m is the slope of the line and b is the y-intercept of the line.

Example:

Example:

m = y2 - y1/x2 - x1
= -1(--5)/ 6-0
= -1+5/ 6
= 4/6
= 2/3
The slope is 2/3

The y-intercept is -5

Equation =y = mx + b
y = 2/3x + (-5)
The equation of the line is y = 2/3x + (-5)

A horizontal line is written in the form y = b, where b is the y-intercept.

A horizontal line is written in the form
y = b, where b is the y-intercept.

m = 4-4/ 2-0 = 0/2 =0 The slope is 0 The y-intercept is 4 y= 0x +4 y= 4

m = 4-4/ 2-0
= 0/2
=0
The slope is 0
The y-intercept is 4
y= 0x +4
y= 4

A vertical line is written in the form x = a, where a is the x-intercept. The slope of a vertical line is undefined.

A vertical line is written in the form
x = a, where a is the x-intercept. The slope of a
vertical line is undefined.

m = 1 - 0/ 0-0 = 1/0 Since division by zero is unidentified and there is no y-intercept, the equation for this line be x

m = 1 - 0/ 0-0
= 1/0
Since division by zero is unidentified and there is no y-intercept, the equation for this line be x = 2

To graph a line when given m and b,

y = mx + b = 3/4x + (-2) Start off by plotting -2 first. Then move 4 to the right and up 3 to find the second point. Rep

y = mx + b
= 3/4x + (-2)
Start off by plotting -2 first. Then move 4 to the right and up 3 to find the second point. Repeat this process until a longer line can be made.

The y-intercept is -2. The slope is 3/4

Graph a Line Using Intercepts

The x-intercept is the x-coordinate of the point where a line crosses the x-axis. At this point, y=0. The y-intercept is the

The x-intercept is the x-coordinate of the point where a line crosses the x-axis. At this point, y=0.
The y-intercept is the y-coordinate of the point where a line crosses the y-axis.
At this point, x=0.

y-intercept 3x - 2y - 6 = 0 Let y = 0 3x - 2(o) - 6 = 0 3x - 6 +6= 0 +6 3x/3 = 6/3 x = 2 x-intercept 3x - 2y - 6 = 0 Let x

y-intercept
3x - 2y - 6 = 0
Let y = 0
3x - 2(o) - 6 = 0
3x - 6 +6= 0 +6
3x/3 = 6/3
x = 2

x-intercept
3x - 2y - 6 = 0
Let x = 0
3(0) - 2y - 6 = 0
-2y - 6 +6 = 0 +6
-2y/2 = 6/-2
y = -3

Graph

Graph

Note: Graphing using the intercept method may not be the best method to choose when the values that we are working with in an equation are not divisible by the coefficient by which we end up working with equations

Find slope using the intercepts

If the x-intercept is -4 and the y-intercept is -6, you would substitute these points in the formula. m = y1-y2/x1-x2
= -6 -6/0- (-4)
= -6/4
= -3/2
The slope of the line -3/2

Parallel and Perpendicular lines

Parallel Lines: Lines that run in the same direction and never cross. Matching arrow symbols are used to indicate that the li

Parallel Lines:
Lines that run in the same direction and never cross. Matching arrow symbols are used to indicate that the lines are parallel. The slopes are the same

The equation of the line is y = 3x -4. Give the slope of the parallel line. The line y = 3x -4 has the slope 3. A para

The equation of the line is y = 3x -4. Give the slope of the parallel line. The line y = 3x -4 has the slope 3.
A parallel line will have the same slope, 3.

Perpendicular Lines: Lines that intersect at right angles – i.e. 90°. A small box is often used to indicate that the angle be

Perpendicular Lines:
Lines that intersect at right angles – i.e. 90°. A small box is often used to indicate that the angle between the two lines is 90°. The slopes are negative reciprocals. Negative reciprocals are 2 numbers whose product is -1

the equation of a line is y = 3/5x + 2. Give the slope pf the perpendicular line. The line y = 3/5x + 2 has the slope ⅗. The

the equation of a line is y = 3/5x + 2. Give the slope pf the perpendicular line. The line y = 3/5x + 2 has the slope ⅗.
The perpendicular line will have slope -5/3.
(Turn the fraction upside down and use opposite operation)

Find an Equation for a Line Given the Slope and a point

To find the equation of a line if you know its slope and one point on the line:
- Substitute the given slope for m and the coordinates of the given point into the equation y= mx+b and solve for b. -Write the equation by substituting the values for m and b into y = mx + b

Slope = 1/2 that passes through the point (1,5)

Slope = 1/2 that passes through the point
(1,5)

y = mx + b
5 = ½ (1) + b
5 = ½ + b
5 - ½ = b
4 ½ = b y= 1/2x+9/2
The y-intercept is 4 ½ or 9/2

Mary knows that it costs $25 to take a taxi to work, which is 10km from her home. She forgets what the fixed cost is, but rem

Mary knows that it costs $25 to take a taxi to work, which is 10km from her home. She forgets what the fixed cost is, but remembers that the variable cost is $2/km. Her friend lives 12km from Mary’s home. Mary has $60 to spend on the weekend. Can she afford a round trip to see her friend?

To Find the fixed cost, substitute d = 10, C = 25, and m = 2 into C = md + b and solve for b

C = md + b
25 = 2(10) + b
25 = 20 + b
25 -20 = b
5 = b
The vertical intercept is 5
The equation C = 2n + 5 gives the cost, C, in dollars, for a trip d kilometres long

Graph

Graph

Find the Equation of a Line given 2 Points

You can find an equation for a line if you know two points on the line.

- Find the slope by substituting the two points into the slope formula

- Find the y-intercept by substituting the slope and one of the two points into y=mx+b , and then solve for b (the choice of which point you choose does not matter).

Write the equation by substituting m and b into y=mx+b

A line that passes through the points (1,2) and (5,10). Find the equation for the line Step 1. find the slope m = 10 -2/5 -1

A line that passes through the points (1,2) and (5,10). Find the equation for the line
Step 1. find the slope
m = 10 -2/5 -1
= 8/4 or 2
Step 2. Find the y-intercept
2 = 2(1) + b
2 = 2 + b
0 = b
Step 3. write the equation
y = (2)x + (0)
= 2x + 0
= 2x

Direct Variation

A relationship between 2 variables in which 1 variable is a constant multiple of the other.

Constant of variation: in a direct the ratio of corresponding values of the variables, often represented by k, or the constant multiple by which one variable is multiplied. In the phrase, "The total cost varies directly with the number of books bought. 5 books cost $35" is a direct variation. So the constant of variation would be 7 because 35 divided by 5 is 7.

if d varies directly as t, then the constant of variation, k, is given by k = d/t or d = kt

if d varies directly as t, then the constant of variation, k, is given by k = d/t or d = kt

The graph's line is a straight line that passes through the origin

The graph's line is a straight line that passes through the origin

Example: The cost of oranges varies directly with total mass bought. 2 kg of oranges costs $4.50. To write the relationship i

Example: The cost of oranges varies directly with total mass bought. 2 kg of oranges costs $4.50. To write the relationship in words you would say,"to get the cost of C, of oranges, multiply the mass r, in kg, of oranges, by $2.25. In this case the constant of variation represents the constant average cost, $2.25/ kg. C = 2.25 r.

Partial Varition

Partial variation is a relationship between 2 variables in which the dependent variable is the sum of a constant number and a constant multiple of the independent variable

A partial variation has an equation of the form y=mx+b, where b represents the fixed, or initial value of y and m represents the constant of variation.

The graph's line is straight line that doesn't pass through the origin.

The graph's line is straight line that doesn't pass through the origin.

Kristina works as a sales representative at the local toys store. She earns a weekly salary of $100 plus 15% commission on he

Kristina works as a sales representative at the local toys store. She earns a weekly salary of $100 plus 15%
commission on her sales.

Table of values

Table of values

Graph

Graph

Initial value: 100. The constant of variation: 0.15

Equation: let E represent earnings and let s represent number of sales. E=0.15s+100

Slope

The steepness of a line is measured by its slope. The slope is the ratio of the rise to the run and is often represented by
the letter m. We use the formula slope = rise/run to calculate slope. The rise is the vertical distance between two points, while the run is the horizontal distance between two points.

Rise shows if you are going up or down and the run tells you if the direction is left or right. A rise of 3 means go up 3, and run of 4 means go right 4.

positive graph

positive graph

Rise=3 Run= 2 slope= 3/2

Negative graph

Negative graph

Rise=-2 Run=3 Slope=-2/3

Use the slope to find a point

If a line segment has one endpoint, A(4,7), and slope of -5/3. To find the coordinates of another possible endpoint, B you ne

If a line segment has one endpoint, A(4,7), and slope of -5/3. To find the coordinates of another possible endpoint, B you need to 3 right and 5 down. so the answer could be B(7,2).

Horizontal Slope

Horizontal Slope

rise=0 run=4

m=0/4. Slope is zero.

Vertical Slope

Vertical Slope

rise=-4 run=0

The slope is unidentified

For safety, the slope of a staircase must be greater than 0.58 and less than 0.70. A staircase has a vertical rise of 2.5m ov

For safety, the slope of a staircase must be greater than 0.58 and less than 0.70. A staircase has a vertical rise of 2.5m over a horizontal run of 3.7m. Is the staircase safe?

m=rise/run
=2.5/3.7
= 0.68

Slope as a Rate of Change

Rate of change is the changes in one quantity relative to the change in another.

Alexa runs every morning before school. One day she ran 5 km in 20 min. To calculate the rate of change you need to divide th

Alexa runs every morning before school. One day she ran 5 km in 20 min. To calculate the rate of change you need to divide the change in distance by change in time. So it would be 5/20 = 0.25. Therefore the rate of change is 0.25 km/min.

Rate of change requires units, such as kilometres per hour.

0.25 km/min

0.25 km/min

When a relation is graphed, the slope describes the rate of change.

Example:

Example:

To find the slope of a line segment joining 2 points, subtract the y-values to get the rise in the same order to get the run.

Example:

Example:

m = rise/run
= 5 - 65 /500 - 0
=-60/500
=-0.12

The rate of change is -0.12/km. The car uses 0.12 litre of gasoline per km driven. The rate of change is negative since the gasoline is always decreasing.

First Differences

Differences between consecutive y-values with evenly spaced x-values

To work with first differences, the values of x (independent variable) must be change by a constant amount.

To work with first differences, the values of x (independent variable) must be change by a constant amount.

To find first differences, subtract consecutive values of y (dependent variale)

To find first differences, subtract consecutive values of y (dependent variale)

If the differences of a relation are constant, the relation is linear

If the differences of a relation are not constant, the relation is non-linear

Connecting Variation, Slope, and first differences

The slope of a linear relation remains constant. The first differences also remain constant when the changes in the x-values are constant.

Since the slope of a linear relation is constant, you can use any pair of points and the slope will be the same.

The slope, m, of a line can be calculated by dividing the change in y by the change in x.

Slope can be symbolised as m= delta-y/delta-x, where delta represents change in.

Slope can be symbolised as m= delta-y/delta-x, where delta represents change in.

Algebraically, The equation of a line has the form y = mx + b, where m represents the slope and b represents the vertical intercept, or the value of the dependent variable where the line intersects the vertical or y-axis.

A mathematical relation can be described in four different ways:
- Using words - Using a diagram or graph - Using numbers - Using an equation

Words: Abby earns $25 per day plus $9 per 100 flyers foe delivering newspapers.

Numbers:

Numbers:

Graph

Graph

Equation: E = 9/100F + 25, where E is Abby's earnings in dollars, and F is he numbers of flyers delivered. The slope is Abby's rate of pay in relation to the number of flyers delivered.